On the use of super compact scheme for spatial differencing in numerical models of the atmosphere

The 'Super Compact Finite-Difference Method' (SCFDM) is applied to spatial differencing of some prototype linear and nonlinear geophysical fluid dynamics problems. An alternative form of the SCFDM relations for spatial derivatives is derived. The sixth-order SCFDM is compared in detail with the conventional fourth-order compact and the second-order centred differencing. For the frequency of linear inertia-gravity waves on different numerical grids (Arakawa's A-E and Randall's Z) related to the Rossby adjustment process, the sixth-order SCFDM shows a substantial improvement on the conventional methods. For the Jacobians involved in vorticity advection by non-divergent flow and in the Bolin-Charney balance equation, a general framework, valid for every finite-difference method, is derived to present the discrete forms of the Jacobians. It is found that the sixth-order SCFDM provides a noticeably more accurate representation of the wave-number distribution of the Jacobians, when compared with the conventional methods. The problem of reconstructing the stream-function field from the vorticity field on a sphere is also considered. For the Rossby-Haurwitz wave, the computation of a normalized global error at different horizontal resolutions in longitude and latitude directions shows that the sixth-order SCFDM can markedly improve on the fourth-order compact. The sixth-order SCFDM is thus proposed as a viable method to improve the accuracy of finite-difference models of the atmosphere.